Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)
Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)
Microsoft PowerPoint - Lecture Notes 1 - 213 (Oscillations)
(Synonym:Vibration)
What means Oscillation? Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. Many things oscillate/vibrate: Periodic motion (a motion that repeats itself over and over)
Pulse Oscillations are the origin of the sensation of musical tone
.. in Aerospace: Orbits Electrical/Computer: LRC resonance in circuits Physics: Atomic Vibrations, String Theory, Electromagnetic Waves Why does something vibrate/oscillate? Whenever the system is displaced from equilibrium, a restoring force pulls it back, but it overshoots the equilibrium position.
more examples
.heart beat, breathing, sleeping, taking shower, eating, chewing, blinking, drinking .motion of planets, stars, motion of electrons, atoms .wind (Tacoma Bridge) .vocal cords, ear drums
Pendulum
Simple harmonic motion (SHM) occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium. For instance when the restoring force is F = - k x.(Hooks Law)
Period T
2 x m 2 = kx t 2 x k + x=0 2 m t 2 x 2 + x=0 2 t
A is the amplitude of the motion, the maximum displacement from equilibrium, A=vmax, and A2 =amax.
Mass-Spring Java applet
Mass-Spring-System
A Spring always pushes or pulls mass back towards equilibrium position. The time period can be calculated from Hookes Law:
F = kx F = ma ma = kx m[ A 2 cos(t )] = k[ A cos(t )] k m m k
2 =
or T = 2
(Application: measure the mass of astronauts in space) Heavier mass slower oscillations Stiffer spring (greater k) rapid oscillations
1 2 = f
Energy: E = U + K U: Potential Energy K: Kinetic Energy Spring Epot = U = k x2 Ekin = K = m v2 Turning points: E = Umax + 0 (Displacement and U at maximum) Minimum: E = Kmax + 0 (Velocity and K at maximum) Total energy of system E = U + K = k (A cos(t))2 + m (A sin(t))2 = k A2 cos2(t) + m A2 2 sin2(t) E = k A2 cos2(t) + m A2 k/m sin2(t) = k A2 (cos2(t)+ sin2(t)) And therefore: E = k A2
Exercise 1: The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point? 1 2 1 2 1 2 1 2 At equilibrium x=0: E = K + U = mv + kx = kA = mv 2 2 2 2
Since E=constant, at equilibrium (x = 0) the KE must be a maximum. Here v = vmax = A. The amplitude A is given, but is not.
Exercise 2: The diaphragm of a speaker has a mass of 50.0 g and responds to a signal of 2.0 kHz by moving back and forth with an amplitude of 1.810-4 m at that frequency.
(a) What is the maximum force acting on the diaphragm?
2 2 ( ) F = F = ma = m A = mA 2 f = 4 mAf max max 2 2
(b) What is the mechanical energy of the diaphragm? Since mechanical energy is conserved, E = KEmax = Umax.
U max = KEmax
1 2 kA 2 1 2 = mvmax 2
KEmax =
f =
2 2
= 0.250 Hz
vmax = A = (8.00 cm )(1.57 rads/sec) = 12.6 cm/sec amax = A 2 = (8.00 cm )(1.57 rads/sec) = 19.7 cm/sec2
2
xmax = A = 8.00 cm
A torsional pendulum is an oscillator for which the restoring force is torsion. For example, suspending a bar from a thin wire and winding it by an angle , a torsional torque is produced, where is a characteristic property of the wire, known as the torsional constant. Therefore, the equation of motion is where I is the moment of inertia. But this is just a simple harmonic oscillator with equation of motion
where
A mass, called a bob, suspended from a fixed point so that it can swing in an arc determined by its momentum and the force of gravity. The length of a pendulum is the distance from the point of suspension to the center of gravity of the bob. Chance observation of a swinging church lamp led Galileo to find that a pendulum made every swing in the same time, independent of the size of the arc. He used this discovery in measuring time in his astronomical studies. His experiments showed that the longer the pendulum, the longer is the time of its swing.
Pendulum
with (expressed in radian measure). (As an example, if = 5.00 = 0.0873 rad, then sin = 0.0872, a difference of only about 0.1%.) With that approximation and some rearranging, we then have
2 mgL + =0 2 t I
Physical Pendulum, Small amplitude
UPendulum
For smaller displacements, the movement of = mgh = mgL (1-cos) an ideal pendulum can be described mathematically as simple harmonic motion (like the mass-spring), as the change in potential energy at the bottom of a circular arc is nearly proportional to the square of the displacement. Real pendulums do not have infinitesimal displacements, so their behaviour is actually of a non-linear kind.
Simple Pendulum
All the mass of a simple pendulum is concentrated in the mass m of the particle-like bob, which is at radius L from the pivot point. Thus, we can substitute I = mL2 for the rotational inertia of the pendulum.
L T = 2 g
Exercise 4: A clock has a pendulum that performs one full swing every 1.0 sec. The object at the end of the string weights 10.0 N. What is the length of the pendulum?
L T = 2 g
Solving for L:
Pivot
CM
mg
The period is
1 ML2 I 2L 3 T = 2 = 2 = 2 1 gMl 3 g gM ( L) 2
Ring
r CM
Disc
r CM
I 2 Mr 2 T = 2 = 2 gMl gMr 2r T = 2 g
I = 2 T = 2 gMl 3r T = 2 2g
3 Mr 2 2 gMr